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https://github.com/hrkrshnn/thurston

Program to compute the space P(n; 3, 3, 3, 3) and plot the elements in H^2
https://github.com/hrkrshnn/thurston

geometry math moduli-space triangulations

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Program to compute the space P(n; 3, 3, 3, 3) and plot the elements in H^2

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* Thurston

#+CAPTION: P(246, 3, 3, 3, 3): The space of all triangulations of the sphere with 2*246 triangles

#+NAME:   fig:p246

[[./246.png]]



  The program does computations of the space P(n; 3, 3, 3, 3), i.e, given 2*n

  equilateral triangles, the space of all different Tetrahedras that can be

  formed by gluing them together. This was written as part of my Master's thesis

  in Mathematics titled "Triangulations of the Sphere after William P. Thurston"

  at Techniche Universität Berlin.

** Requirements

  To compile the code in a Unix/Linux environment, you can use make. The program

  needs [[https://www.boost.org/][Boost C++ libraries]] and [[https://www.mpfr.org/][GNU MPFR]].

** Files

   1. matrix.hpp: contains the data structure to represent the 2x2 matrix.

   2. algorithms.hpp: contains the implementation of `fundTransform`, an

      algorithm that takes a 2x2 integer matrix and transforms it using the

      operators S and T, in such a way that the resulting matrix's represenation

      in Complex plane will lie on the standard fundamental domain of PSL_2(Z).



      genSpace: this is the main algorithm discussed in the thesis. The

      algorithm runs the fundTransform algorithm on a finite set of matrices,

      which can be proven to generate an exhaustive list of matrices

      corresponding to P(n; 3, 3, 3, 3).



      genPoints: this runs the fundTransform algorithm on 'random' matrices of

      determinant $2n$. Given a large enough sample size, in practice, the

      algorithm seems to exhaust the space P(n; 3, 3, 3, 3).

   3. main.cpp. parses the command line arguments and calls the required

      function. The program currently implements the main algorithm from the

      thesis (algorithm 2.) If you need to run the Monte-Carlo style algorithm

      (slower and crude), then change the calls in this file, see comments

      inside the main function.

** R Scripts for visualization

   Some R scripts are included to help visualize the output of the C++ program.

   These scripts can be found under the folder: r-scr.



   The file ~poincare.R~ provides a small R library that can help plot the

   Poincare Upper half space. This *requires* the R package named "plotrix". This

   can be installed by using the command ~install.packages("plotrix")~



   The file ~246.R~ provides the code to visualize the space of P(246; 3, 3, 3,

   3). This requires the poincare.R library.



   *Example*: To see this visualization, inside the r-scr directory do the following:

   #+BEGIN_SRC bash

   R

   > source("246.R")

   #+END_SRC

** Execution

   After running make, the binary will be generated inside the bin folder



   #+BEGIN_SRC txt

   >./thurston --help

   Allowed options:

     --help                      produce help message

     -t [ --triangles ] arg (=6) The number of triangles/2

     -w [ --writefile ]          Writes to file n.txt inside the out folder, where

                                 2*n is the number of triangles

   #+END_SRC



   Example:

   #+BEGIN_SRC txt

     > ./thurston -t 16

     Number of triangles: 32

     [1, 0], [0, 8] z:(4,6.9282)

     [1, 1], [0, 8] z:(4,2.3094)

     [1, 2], [0, 8] z:(2.85714,0.989743)

     [1, 3], [0, 8] z:(2.15385,0.532939)

     [1, 4], [0, 8] z:(1.71429,0.329914)

     [1, 5], [0, 8] z:(1.41935,0.22349)

     [1, 6], [0, 8] z:(1.2093,0.161121)

     [1, 7], [0, 8] z:(1.05263,0.121547)

     [2, 0], [0, 4] z:(1,1.73205)

     [2, 1], [0, 4] z:(1.14286,0.989743)

     [2, 2], [0, 4] z:(1,0.57735)

     [2, 3], [0, 4] z:(0.842105,0.364642)

     [4, 0], [0, 2] z:(0.25,0.433013)

     [4, 1], [0, 2] z:(0.285714,0.329914)

     [8, 0], [0, 1] z:(0.0625,0.108253)



     After transformation, there are 15 elements 

     [1, 0], [-4, 8] z:(0,6.9282)

     [1, 1], [-4, 4] z:(-1.15875e-16,2.3094)

     [1, 2], [-3, 2] z:(-0.142857,0.989743)

     [2, -2], [1, 3] z:(-0.5,1.73205)

     [2, 0], [-3, 4] z:(-0.5,1.73205)

     [1, -3], [3, -1] z:(0.142857,0.989743)

     [1, -2], [4, 0] z:(0,2.3094)

     [1, -1], [4, 4] z:(3.47626e-16,6.9282)

     [2, 0], [-2, 4] z:(0,1.73205)

     [2, 1], [-2, 3] z:(0.142857,0.989743)

     [2, -2], [2, 2] z:(8.69064e-17,1.73205)

     [-2, 1], [0, -4] z:(1.15875e-16,2.3094)

     [0, -2], [4, -2] z:(-8.69064e-17,1.73205)

     [0, -2], [4, -3] z:(0.5,1.73205)

     [0, -1], [8, -4] z:(-3.47626e-16,6.9282)



     After removing duplicates, there are 5 elements.

     [1, 2], [-3, 2] z:(-0.142857,0.989743)

     [2, 0], [-2, 4] z:(0,1.73205)

     [2, -2], [1, 3] z:(-0.5,1.73205)

     [1, 1], [-4, 4] z:(-1.15875e-16,2.3094)

     [1, 0], [-4, 8] z:(0,6.9282)

   #+END_SRC



   The notation "[a b], [c, d]" denotes a 2x2 matrix with first row [a, b] and

   second row [c, d]. The complex number next to it represents the point in the

   upper half plane of the complex plane corresponding to the matrix.

*** Developer notes

    The program can be compiled in debug mode by using `make debug.` This will

    verify some asserts, mostly invariants when the program is executing.